Experimental and Curvature-Based Analysis for Accuracy Optimization in 3-Axis CNC Milling of Concave Parabolic Freeform Surfaces

Introduction. Concave parabolic freeform surfaces are integral to high-performance optical reflectors and precision mechanical components, where stringent geometric accuracy is paramount to functional efficiency. Despite their widespread application, achieving high surface fidelity in 3-axis CNC milling remains a significant technical challenge. This difficulty arises from the intricate, non-linear interactions between cutting tool geometry, machining parameters, and the varying local curvature of the surface. While conventional research often focuses on global parameter optimization, there exists a notable scientific gap in understanding how localized curvature gradients dictate error distribution during the material removal process. This study addresses this gap by establishing a curvature-aware analytical framework aimed at identifying the fundamental drivers of machining inaccuracies in parabolic geometries. Materials and Methods. The research methodology integrates theoretical modeling with rigorous experimental validation. Initially, a mathematical model based on principal deviation theory was formulated to characterize the geometric deviations inherent in the 3-axis milling process. This analytical foundation allowed for the mapping of theoretical errors against surface differential geometry. Subsequently, an experimental investigation was executed using a Taguchi L 9 orthogonal array design to systematically evaluate the influence of three primary machining factors: tool diameter, feed rate, and step-over. Concave parabolic samples were machined and measured using high-precision metrology equipment. The resulting data were processed via Analysis of Variance (ANOVA) and Signal-to-Noise (S/N) ratio analysis to quantify the individual and interactive statistical contributions of each parameter to the total machining error. Results. The findings demonstrate that tool diameter and step-over are the predominant factors, accounting for the majority of the variance in machining accuracy, whereas the influence of feed rate is found to be statistically marginal within the tested range. Crucially, the results have revealed that machining errors are not uniformly distributed but are highly correlated with the local principal curvatures of the parabolic profile. MATLAB simulations further corroborated these findings, showing that the maximum deviation occurs in regions of high curvature where the tool-surface engagement geometry is most constrained. These specific results provide a quantitative map of how tool geometry interacts with parabolic concavity to produce predictable error patterns. Discussion. This research provides a novel interpretation of machining errors through the lens of surface differential geometry, successfully bridging the gap between theoretical surface modeling and practical CNC manufacturing. The findings demonstrate that a “one-size-fits-all” toolpath strategy is inherently insufficient for concave freeform geometries due to localized geometric complexities. Instead, this study validates that the implementation of an adaptive step-over strategy, dynamically adjusted based on local curvature values, significantly enhances surface forming accuracy. The core scientific novelty lies in the proposed integration of curvature-based compensation directly into the toolpath planning phase, allowing for proactive error mitigation rather than reactive adjustments. Conclusion. In conclusion, the proposed approach offers a robust and scalable solution for improving the precision of complex freeform components in real-world industrial environments. By adopting this methodology, high-precision manufacturing processes can potentially reduce post-processing requirements and overall production costs. While this study focuses on static and geometric accuracy, future research will extend this model to incorporate the assessment of dynamic processing errors. This expansion will provide a more comprehensive framework for accuracy optimization in multi-axis CNC machining of complex surface.